On the Optimal Policy Structure in Serial Inventory Systems with Lost Sales

Transcription

1 On he Opimal Policy Srucure in Serial Invenory Sysems wih Los Sales Woonghee Tim Huh, Columbia Universiy Ganesh Janakiraman, New York Universiy May 21, 2008 Revised: July 30, 2008; December 23, 2008 Absrac We sudy a periodically reviewed, serial invenory sysem in which excess demand from exernal cusomers is los. We derive elemenary properies of he vecor of opimal order quaniies in his sysem. In paricular, we derive bounds on he sensiiviy (or, more mahemaically, he derivaive) of he opimal order quaniy a each sage o he vecor of he curren invenory levels. Our analysis uses he concep of L-nauralconvexiy, which was sudied in discree convex analysis (Muroa (2003)) and recenly used by Zipkin (2008) for sudying single-sage invenory sysems wih los sales. We also remark on how our analysis exends o models wih capaciy consrains and/or backordering. Indusrial Engineering and Operaions Research Deparmen, Columbia Universiy, 500 Wes 120h Sree, New York, NY 10027, USA. huh@ieor.columbia.edu. Research suppored parially by NSF gran DMS Sern School of Business, New York Universiy, 44 Wes 4h Sree, New York, NY 10012, USA. gjanakir@sern.nyu.edu. 1

2 1 Inroducion We consider a periodically-reviewed, single-iem invenory sysem wih J sages in series. Excess demand ha canno be saisfied immediaely a he mos downsream (i.e., cusomerfacing) sage is los. Los sales invenory sysems are known o be difficul o analyze, and he opimal policy for even he single-sage sysem is complicaed when he order delivery lead-ime is posiive. (When he lead-ime is zero, he problem becomes he sandard muliperiod newsvendor problem for which order-up-o policies are opimal.) For single-sage sysems, a parial characerizaion of he srucure of he opimal policy has been provided by Karlin and Scarf (1959) and Moron (1969), and heir resuls are recenly reinerpreed and srenghened by Zipkin (2008). For muli-echelon sysems, no such resuls are known; in fac, o our knowledge, his work is he firs aemp o sudy he srucure of opimal policies in muli-echelon sysems wih los sales. In paricular, we provide bounds on he sensiiviy of he opimal order quaniy a each sage wih respec o he vecor of he invenory levels. In his paper, we use he concep of L-naural-convexiy (L -convexiy) from discree convex analysis (Muroa (2003)), recenly used by Zipkin (2008) for single sage sysems, o esablish properies of he opimal policy in serial sysems. Anoher recen example of is use in invenory heory is Lu and Song (2005). We refer he reader o Chaper VII of Fujishige (2005), Muroa (2003), and Zipkin (2008) for a more deailed discussion on his concep. Our analysis is based on wo main ideas. The firs one is o consider J equivalen versions, one for each sage, of he sochasic dynamic program corresponding o he problem of minimizing he expeced coss incurred by his sysem over some finie horizon of periods. The j h version is used o generae he sensiiviy resul relaed o he order quaniy of sage j. The second one is o ransform he sae vecor such ha a sae can be represened by a so-called bi-echelon invenory vecor, wih respec o which he L -convexiy propery can be esablished. To he bes of our knowledge, boh of hese ideas are novel in he invenory lieraure. 2

3 2 Model and Resul We consider a serial invenory sysem wih los sales under periodic review. The sysem consiss of J 1 sages, indexed by j = 0, 1,..., J 1. The lowes sage facing exogenous demand is represened by sage 0, sage j orders from sage j + 1 where j {0,..., J 2}, and sage J 1 orders from an ouside supplier wih infinie supply. The replenishmen leadime from one sage o anoher is deerminisic, and we assume iniially ha each lead-ime is 1 period. (In Secion 4, we commen on how our analysis exends o he case of arbirary lead imes.) In each period, he following sequence of evens occurs: (1) receip of delivery a every sage, (2) order placemen a every sage, and (3) demand realizaion. The demand in each period is saisfied o he exen possible, and any demand ha canno be saisfied immediaely is los. Le p 0 represen he per-uni los sales penaly cos, and le H j 0 denoe he holding cos a sage j, ha is, he cos for holding one uni a sage j for a period. These coss are charged a he end of a period. For any j {0, 1,..., J 1}, le x j 0 denoe he sage-j invenory level afer receiving deliveries a he beginning of a period, and le q j [0, x j+1 ], where x J is aken o be, denoe he quaniy ordered by sage j in a period. Thus, he vecor (x 0, x 1,..., x J 1 ) represens he sae a he beginning of a period afer receiving deliveries, and he vecor ( q 0, q 1,..., q J 1 ) represens he acion in a period. We use x and q o denoe he vecors (x 0,..., x J 1 ), and ( q 0,..., q J 1 ), respecively. Le D denoe he demand in period. We assume ha demands are independenly disribued across periods. Nex, we define he dynamic program for his sysem for a planning horizon of T periods indexed by = 1, 2,..., T. Periods are indexed forwards, i.e., period + 1 follows period. Given a sae-acion pair of x and q in period, he sage-j invenory level in period + 1 is given by (x 0 D ) + + q 0 if j = 0 x j q j 1 + q j if j {1,..., J 1}. 3

4 Le f T +1 (x) = 0 for all x. Consider any {1,..., T }. Le γ (0, 1] denoe he discoun facor for capuring he ime value of money. Define [ ] J 1 f (x) = E H 0 (x 0 D ) + + H j x j + p (D x 0 ) + + γ min q j=1 [ ( E f+1 (x 0 D ) + + q 0, x 1 q 0 + q 1,..., x J 1 q J 2 + q J 1)] s.. 0 q j x j+1, j = 0,..., J 1. Le q (x) denoe a minimizing vecor, q, in he opimizaion problem above. Thus, q j denoes he order quaniy for sage j, according o his vecor. [Noe: When he opimizaion problem above does no have a unique soluion, any saemens we make on properies of q should be aken o mean he exisence of an opimal selecor q (x) (precisely defined in Secion 3.4) wih hose properies.] We are now ready o sae he main resul of he paper. Theorem 1. Assume ha q is a differeniable funcion of x. Then, he following inequaliies hold: If q 1 q q 0 for every k {0,..., J 1}, x k x 0 0 q q x J 1 x k+1 1 for every k {0,..., J 2}, and q q xk+1 x k 1 for every k {0,..., J 2}. is no differeniable, hen he inequaliies above hold afer replacing every quaniy of he form q x j wih q (x+ j ) q (x), where j is a vecor wih in is j h componen and zero in all he oher componens and is any sricly posiive number. In words, Theorem 1 saes ha he opimal order quaniy a sage k is a decreasing funcion of he invenory a any downsream sage j (i.e., j k), and ha he rae of his decrease is smaller han 1. On he oher hand, he opimal order quaniy a sage k is an increasing funcion of he invenory a any upsream sage j (i.e., j > k), and he rae of his increase is also smaller han 1. Furhermore, he sensiiviy of he opimal order quaniy a a sage o he invenory level a a downsream (upsream) sage is greaer he 4

5 less downsream (upsream) he laer sage is relaive o he former sage. Finally, he rae of decrease of he opimal order quaniy a a sage wih respec o any downsream sage s invenory plus he rae of is increase wih respec o any upsream sage s invenory is iself smaller han 1. We presen he proof of his heorem in Secion 3. 3 Proof of Theorem Preliminary Resuls Le V R n be a polyhedron ha forms a laice. A funcion f : V R is submodular provided ha f(x) + f(y) f(x y) + f(x y), for any pair of x and y in V, where and are he componen-wise minimum and componen-wise maximum operaors. Le e = (1, 1,..., 1) be he vecor of 1 s wih an appropriae lengh. Following Zipkin (2008), we say ha f : V R is L -convex if ψ(v, ζ) = f(v ζe), ζ 0, is submodular on {(v, ζ) v V, ζ R, v ζe V }. The following basic properies of L -convexiy are sraighforward exensions of resuls presened in Zipkin s paper. Lemma 2. (a) If f(v) is L -convex, hen ψ(v, ζ) = f(v ζe) is also L -convex. (b) Le r j and u j be fixed for all j {1,..., m} such ha r j u j, and le ξ ij and ς ij be fixed for all (i, j) {1,..., n} {1,..., m} such ha ς ij ξ ij. Define S = { (v, w) R n R m ς i,j v i w j ξ i,j (i, j), and r j w j u j j }. Suppose ha g(v, w) is a L -convex funcion defined on S. Le f(v) = min w {g(v, w) (v, w) S}. Then, f is L -convex. 5

6 (c) Le I be a subse of {1,..., n}, and suppose ha g(v, ζ) is L -convex, defined on {v : v i 0 for all i I}. Le ζ(v) denoe he larges value of ζ ha solves min ζ { g(v, ζ) v i ζ 0 for i I }. Then, ζ(v) is nondecreasing in v, and ζ(v + ωe) ζ(v) + ω for ω > 0. Proof. While par (a) is due o Zipkin (2008), pars (b) and (c) require minor modificaions. We will firs prove par (b), which slighly generalizes Lemma 2 of Zipkin (2008). Le ψ(v, ζ) = f(v ζe) = { min g(v ζe, w) ς i,j v i ζ w j ξ i,j (i, j), r j w i u j j } w = min w { g(v ζe, w ζe) ς i,j v i w j ξ i,j (i, j), r j w i ζ u j j }. Since each consrain in he above minimizaion problem has exacly wo non-zero variables of (v, w, ζ) and hese wo variables have he opposie signs, he feasible region forms a laice (Topkis (1998), Example 2.2.7). Thus, ψ(v, ζ) is submodular (Topkis (1998), Theorem 2.7.6), and we conclude ha f is L -convex. We will now prove par (c). Since g is submodular and he feasible region of he minimizaion problem forms a laice, ζ(v) is nondecreasing in v by Theorem of Topkis (1998). Also, for any ω > 0 and ζ 0 saisfying ζ > ζ(v) + ω, g(v + ωe, ζ) g(v + ωe, ζ(v) + ω) g(v, ζ ω) g(v, ζ(v)) > 0. (1) The firs inequaliy in (1) is rue because of he following argumen. 1 Define ψ(v, ζ, ζ ) = g(v ζ e, ζ ζ ). Since g is L convex, we know ha ψ is submodular in (v, ζ, ζ ); in paricular, for a given v, ψ(v, ζ, ζ ) is submodular in (ζ, ζ ). This implies ha ψ(v, ζ ω, ω) ψ(v, ζ(v), ω) ψ(v, ζ ω, 0) ψ(v, ζ(v), 0). 1 An alernae proof of his inequaliy is based on he following propery known as ranslaionsubmodulariy (see Theorems 7.2 and 7.29 of Muroa (2003) and he paper by Muroa and Shioura (2004)): if g is L convex, hen g(p) + g(q) g((p α e) q) + g(p (q + α e)) for all α 0. The desired inequaliy follows by choosing p = (v+ωe, ζ), q = (v, ζ(v)) and α = ω and from he facs ha (p α e) = (v, ζ ω) q and (q + α e) = (v + ωe, ζ(v) + ω) p. 6

7 Observe ha he lef and righ sides of he inequaliy above are equal o he lef and middle expressions of (1), respecively. This complees he proof of he firs inequaliy in (1). The second inequaliy in (1) follows from he assumpion ha ζ ω > ζ(v) and he fac ha, for every ζ > ζ(v), he inequaliy g(v, ζ ) > g(v, ζ(v)) holds due o he definiion of ζ(v). Now, from (1), we obain g(v + ωe, ζ) > g(v + ωe, ζ(v) + ω), which implies ha ζ canno be opimal for v + ωe. Thus, ζ(v + ωe) ζ(v) + ω. 3.2 Sae Transformaion We fix k {0..., J 1}, one of he sages of he sysem, whose opimal order quaniy we will analyze, and inroduce an alernae represenaion of he sae. Define w j = x j + x j x k for j {0, 1,..., k}, and, (2) u j = (x k+1 + x k x j ) for j {k + 1,..., J 1} and k J 2. (3) Le w = (w 0,..., w k ). For k J 2, le u = (u k+1,..., u J 1 ). (For conciseness, we refrain from saing k J 2 each ime u is used. If k = J 1, u should be reaed as he null vecor.) We refer o (w, u) as he bi-echelon invenory vecor anchored a k. Then, (w, u) is a valid represenaion of he sae, where he corresponding sae space is given by V = { (w, u) w 0 w k 0 u k+1... u J 1}. Noice ha V is a laice. For our analysis, i is convenien o consider he following modificaions o our model. Firs, we divide he se of ordering decisions ino wo seps, he firs one for sage k and he second one for all he oher sages. Second, we no longer insis ha he manager mus saisfy demand o he maximum exen possible; we give he manager he flexibiliy of selecing he sales quaniy (provided ha he sales quaniy does no exceed he demand and does no exceed he available invenory a sage 0). I is easy o show ha i is opimal o saisfy demand o he maximum exen possible. Of hese wo modificaions, he firs is a echnique which is new o he muli-echelon invenory lieraure whereas he second appears in Zipkin (2008). Now, he revised sequence of evens is as follows: (1) receip of delivery a all sages, 7

8 (2a) order placemen for sage k, (2b) order placemen for all he oher sages, (3) demand realizaion, and (4) he sales quaniy decision. We emphasize ha he modified problem remains equivalen o he original problem, and hese modificaions are inroduced for he convenience of our analysis only. Le q j 0 denoe he quaniy ordered by sage j. For j < J 1, his quaniy q j is consrained above by he amoun of available invenory in he immediaely upsream sage, x j+1. Define w j + = w j q j 1 for j {1,..., k}, and u j + = u j q j for j {k + 1, k + 2,..., J 1}. Le ŵ + = (w 1 +,..., w k +) and u + = (u k+1 +,..., u J 1 + ). Then, (w 0, ŵ +, u + ) corresponds o he sae in he nex period, assuming ha sage k does no order in he curren period, and he realized demand is zero. Furher, le ζ 0 denoe he negaive of he order quaniy of sage k in he curren period, i.e., ζ = q k. I follows ha (ζ, ŵ +, u + ) could be used o represen he order placemen acion in he curren period; ha is, given (w, u) and (ζ, ŵ +, u + ), he vecor of order quaniies a all sages is compleely deermined. The feasible order placemen acion (ζ, ŵ +, u + ) corresponding o he sae (w, u) can be characerized by he following se of consrains: w j+1 w j + w j for j {1,..., k} (4) u j u j + u j+1 for j {k + 1, k + 2,..., J 1} (5) where we le w k+1 = 0, and u J =. 0 ζ u k+1, (6) The sales decision is made afer demand is realized. Le d denoe he realized demand. Given he curren sae (w, u) and he order placemen acion (ζ, ŵ +, u + ) in he curren period, he sales quaniy can be deduced from he sae of he nex period. Le w 0 + denoe he firs componen of he sae a he end of he curren period afer sales have occurred; hen, he sales quaniy is w 0 w 0 +. The se of feasible values for w 0 + is given by w 0 d w 0 + and w 1 w 0 + w 0, (7) 8

9 since he sales quaniy is non-negaive and can neiher exceed demand nor he invenory a sage 0, ha is, w 0 w 1. We now commen on he ransiion of saes from one period o he nex. Given he bi-echelon sae vecor (w, u) and he acion vecor (ζ, ŵ +, u +, w+), 0 he order quaniy ζ a sage k shifs he corresponding amoun of invenory from sage k + 1 o sage k. Thus, in he nex period, each w j increases by ζ for j {0, 1,..., k} and each u j also increases by ζ for j {k + 1, k + 2,..., J 1}. (See he definiions of w j and u j in (2) and (3).) Le w + = (w+, 0..., w+) k = (w+, 0 ŵ + ). Then, he sae vecor in he nex period is given by (w +, u + ) ζe. 3.3 L -Convexiy Properies Le us now formally define he T -period sochasic dynamic program for our sysem using he sae vecor (w, u) and he acions (ζ, ŵ +, u +, w+). 0 For any (w, u), le f T +1 (w, u) = 0. Fix {1, 2,..., T }. Define, for every (w, u, ζ, w +, u + ) and d 0 saisfying consrains (4)-(7), ψ (w, u, ζ, w +, u + d) = H 0 (w 0 + w 1 ) + k H j (w j w j+1 ) + j=1 J 1 j=k+1 + p (w 0 + w 0 + d) + γ f +1 ((w +, u + ) ζe), H j ( u j + u j 1 ) where w k+1 = 0 and u k = 0. (The equaion above can be undersood by observing ha he invenory on hand a sage 0 a he end of he period is w+ 0 w 1 ; he invenory on hand a sage j is (w j w j+1 ) if j {1,..., k} and ( u j + u j 1 ) if j {k + 1, k + 2,..., J 1}; and he amoun of sales los is d (w 0 w+) 0 if d is he demand.) Now, recall ha ŵ + = (w+, 1..., w+) k and w + = (w+, 0 w+, 1..., w+). k Le ˆψ (w, u, ζ, ŵ +, u + d) be he minimum value of ψ opimized over he sales decision w+, 0 i.e., ˆψ (w, u, ζ, ŵ +, u + d) = min w + 0 ψ (w, u, ζ, w +, u + d) subjec o (7). 9

10 Also, le [ ] φ (w, u, ζ, ŵ +, u + ) = E D ˆψ (w, u, ζ, ŵ +, u + D ), g (w, u, ζ) = min ŵ +,u + φ (w, u, ζ, ŵ +, u + ) subjec o (4) and (5), and f (w, u) = min ζ g (w, u, ζ) subjec o (6). Theorem 3. For each {1,..., T }, he funcions g (w, u, ζ) and f (w, u) are L -convex. Proof. Noe ha f T +1 = 0 is L -convex. We assume, as an inducive hypohesis, ha f +1 is L -convex. We will firs prove ha ψ is L -convex. In he definiion of ψ, each of he firs four erms is separable and linear, and is herefore L -convex. In he las erm, since f +1 is L -convex, Lemma 2 implies ha f +1 ((w +, u + ) ζe) is L -convex in (w +, u +, ζ). Therefore, ψ (w, u, ζ, w +, u + d) is a sum of L -convex funcions and is, herefore, L -convex in (w, u, ζ, w +, u + ). Since ψ (w, u, ζ, w +, u + d) is L -convex, Lemma 2(b) implies ha ˆψ (w, u, ζ, ŵ +, u + d) is L -convex for each d. Since he expecaion operaor preserves L -convexiy, i follows ha φ (w, u, ζ, ŵ +, u + ) is L -convex. Again by Lemma 2(b), we obain ha boh g (w, u, ζ) and f (w, u) are L -convex, hus compleing he inducion. 3.4 Implicaions of L -Convexiy Le ζ (w, u) denoe he larges minimizer in he definiion of f for some. Le q (w, u) = ζ (w, u); hus, q (w, u) is an opimal order quaniy for sage k. Assuming ha q (w, u) is differeniable, he following inequaliies are implied by Lemma 2(c): k j=0 q w + J 1 j j=k+1 q w j 0 for j {0,..., k}. (8) q u j 0 for j {k + 1,..., J 1} if k J 2. (9) q u j 1, (10) where he second summaion in he las inequaliy above does no exis if k = J 1. Noice ha Theorem 1 is saed in erms of q, which is a funcion of x, whereas he inequaliies above are saed in erms of q which is a funcion of (w, u). Since q 10 has no

11 been uniquely defined ye, we define q (x) = q (w, u), where w and u depend on x hrough definiions (2) and (3). The resuls of he heorem follow direcly by applying he chain rule for differeniaion and he inequaliies in (8)-(10). The case in which q (x) is no differeniable corresponds o he case in which q is no differeniable. The deails of he proof in his case are provided in he appendix. 4 Remarks In his secion, we remark on how Theorem 1 and our analysis exend o oher seings. (i) Arbirary Lead Times: If he lead imes beween wo successive sages are arbirary inegers, hen we can inser a sufficien number of dummy sages beween hese wo sages o ensure ha he lead ime beween any wo consecuive sages (including he dummy sages) is one period. We hen make sure ha no invenory is held a a dummy sage j by insering an addiional consrain w j + = w j+1 o (4) and u j 1 + = u j o (5). Oherwise, he analysis remains he same and he conclusions of Theorem 1 hold. (ii) Ordering Capaciy: If he quaniy ordered by sage j is consrained above by a capaciy limi CAP j, hen we can add he consrain w j+1 w j+1 + CAP j o (4) and he consrain u j u j + CAP j o (5). The res of he analysis remains unchanged and he conclusions of Theorem 1 hold. (iii) Backorder Models: If excess demand is backordered insead of being los, hen he only required change o our analysis is replacing (7) wih w 0 d = w 0 +. In he backorder sysem wihou capaciy consrains, he conclusion of Theorem 1 rivially follows from he fac ha he opimal policy is an echelon order-up-o policy (Clark and Scarf, 1960). However, when capaciy consrains are presen, hen he srucure of he opimal policy is no known, in general, and our resul here (Theorem 1) is he firs resul which parially characerizes he opimal policy for such sysems. (Parker and Kapuscinski (2004) and Janakiraman and Mucksad (2008) derive he opimal policy 11

12 srucure for such sysems bu hey require he assumpion ha he capaciy limis a all sages are idenical.) We remark ha, in addiion o he firs wo exensions menioned above, Zipkin (2008) also considers sysems in which demand is Markov modulaed and sysems in which here are muliple demand classes; he exensions of our analysis o hese cases are idenical o Zipkin s. Moreover, he sudies he case of sochasic lead imes; our analysis here however does no readily exend o his case. Acknowledgemen We sincerely hank he reviewers, he Associae Edior and he Area Edior for heir commens. In paricular, we are graeful o one of he reviewers who suggesed he alernae proof provided in Foonoe 1 and appropriae references in discree convexiy. References Clark, A., and H. Scarf Opimal Policies for a Muliechelon Invenory Problem. Managemen Science 6: Fujishige, S Submodular Funcions and Opimizaion. Second ed, Volume 58 of Annals of Discree Mahemaics. Amserdam, The Neherlands: Elsevier Science. Janakiraman, G., and J. Mucksad A Decomposiion Approach for a Class of Capaciaed Serial Sysems. Forhcoming, Operaions Research. Karlin, S., and H. Scarf Invenory Models of he Arrow-Harris-Marschak Type wih Time Lag. In Sudies in he Mahemaical Theory of Invenory and Producion, ed. J. Arrow, S. Karlin, and H. Scarf, Chaper 9, Lu, Y., and J. Song Order-Based Cos Opimizaion in Assemble-o-Order Sysems. Operaions Research 53 (1): Moron, T Bounds on he Soluion of he Lagged Opimal Invenory Equaion wih no Demand Backlogging and Proporional Coss. SIAM Review 11 (4):

13 Muroa, K Discree Convex Analysis, Volume 10 of SIAM Monographs on Discree Mahemaics and Applicaions. Philadelphia, PA, USA: Sociey for Indusrial and Applied Mahemaics. Muroa, K., and A. Shioura Fundamenal Properies of M-Convex and L-Convex Funcions in Coninuous Variables. IEICE Transacions on Fundamenals of Elecronics, Communicaions and Compuer Sciences 87 (5): Parker, R., and R. Kapuscinski. 2004, Sep.-Oc.. Opimal Policies for a Capaciaed Two- Echelon Invenory Sysem. Operaions Research 52 (5): Topkis, D. M Supermodulariy and Complemenariy. Princeon, New Jersey: Princeon Universiy Press. Zipkin, P On he Srucure of Los-Sales Invenory Models. Operaions Research 56 (4):

14 A Deails for he Proof of Theorem 1: The Case where q Differeniable is no Recall ha f (w, u) = min ζ g (w, u, ζ) subjec o (6), ha ζ (w, u) is he larges minimizer in his definiion, and q (w, u) = ζ (w, u). We know from Theorem 3 ha g (w, u, ζ) is L -convex. Therefore, Lemma 2 (c) implies ha where e = (1, 1,..., 1) is he vecor of 1 s. q (w, u) is nonincreasing in (w, u), and (11) q ((w, u) + e) q (w, u) for > 0, (12) We will show he inequaliies in he saemen of Theorem 1 for he non-differeniable case using (11)-(12) along wih he definiion of he ransformaion from x o (w, u) in (2)- (3). These inequaliies are direcly implied by he following five saemens, which we will prove in he remainder of his secion: ( q ( q ( q ) (x + k+1 ) q (x) ) (x + j ) q (x) ) (x + j ) q (x) q (x + 0 ) q (x) (x + J 1 ) q (x) ) (x + k ) q (x) ) (x + j 1 ) q (x) ) (x + j 1 ) q (x) q ( q ( q ( q 0 k {0,..., J 1} (13) 0 k {0,..., J 2} (14) 1 k {0,..., J 2} (15) 0 (j, k) {1 j k}. (16) 0 (j, k) {k + 2 j J 1}. (17) To prove saemens (13) and (14), observe ha (11) implies q (x + 0 ) q (x) = q ((w, u) + 0 ) q (w, u) 0 and q (x + J 1 ) q (x) = q ((w, u) J 1 ) q (w, u) 0. (Throughou his proof, we le (w, u) be he ransformed vecor corresponding o x.) 14

15 We show saemen (15) by applying (12) as follows. Then, ( q (x + k+1 ) q (x) ) = q (x + k+1 ) q (x + k ) ( q (x + k ) q (x) ) = q ((w, u) k+1 J 1 ) q ((w, u) k ), where he second equaliy follows from (2)-(3), and he inequaliy follows from (12). This complees he proof of saemen (15). Saemen (16) follows from ( q (x + j ) q (x) ) ( q (x + j 1 ) q (x) ) = q ((w, u) j 1 + j ) q ((w, u) j 1 ) 0, where he inequaliy follows from (11). The proof of saemen (17) is similar. 15